Math Is Fun Forum

The same formal record.  If we look for a parameterization not in 2 and in 3 option the problem can be solved quite simply.

For the equation.

If you know any solution

of this equation.  Then the formula for the solution of the equation can be written immediately.

-   any integers.

It is seen that such formulas can be written infinitely many.  If so like to use the well-known decision, it is better to write like this.  This will allow to solve the equation.

It is easy to see that there are solutions for any

. Ask yourself the number

. And place into factors.

Then the coefficient is set to

us so.

Finding all the factors of

you can write a formula for the parameterization of the solution of this equation.

The transition to 3 parameters

-   Allows not to use too much, but to write formulas.

More interesting the other decisions - when the number is positive.

- the number is specified by the problem statement and can be any.

The equation from there.
http://math.stackexchange.com/questions … -variables

For the equation:

You can record such decisions.

Or.

For the equation.

You can record such decisions.

Or.

http://math.stackexchange.com/questions … -both-prim

Show how the General formula allows us to solve this equation.

We write this equation differently.

Will do the replacement. 

Then the equation takes the form.

Now you can use the General formula. http://www.artofproblemsolving.com/comm … 46h1048219

Let the root equal to 1.  This means that using the solutions of the equation Pell.

Knowing the first solution.

You can find the following solution to the formula.

Using the General formula and the solution of the equation Pell.

You can write the solution of the equation in this form.

I am not interested in his opinion and questions!

The task from there.

http://math.stackexchange.com/questions … 18#1471718

For such a system.

Here is the result.

Number 

are selected so that the desired number intact.

2 books of Diophantus task 34 , 35.  For the system of Diophantine equations.

You can record such a decision.

any integers.

Problem is taken from there.
http://math.stackexchange.com/questions … 72#1418072

For the equation:

Use the standard approach of using Pell equations.  If you use solutions of this equation.

Decisions can be recorded.

If this equation Pell.

Then decisions can be recorded.

Or.

For this.

Number 

  can have any sign.

An interesting case of when. 

The formulas can be seen that while the

can be anything. And 

The system of book 2: objectives 24, 25.

The solutions can be written as.

Or so.

For the system of book 2 task 30.

Decisions will be.

One approach is to write the General formula.  For the equation.

The formula looks for simplicity without coprime solutions. 

After substituting numbers 

  to reduce common divisor.

in such a number.

in such a number.

For the system:

The solutions can be written as.

integers.